The potential energy function of a particle of mass m in one-dimensional motion is given b...

The potential energy function of a particle of mass m in one-dimensional motion is given by

and so the force is of the antirestoring type

F(x) = kx

with x = 0 as a position of unstable equilibrium when k is a positive constant. If the initial conditions are t = 0, x = x0, and x = 0, show that the ensuing motion is given by an expo­nential “runaway”

x(t) = x0(eat + e–at)/2

where the constant